Modeling circadian clocks: From equations to oscillations

Tuning limit cycles with a noise: Survival and collapse

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equation of motion for slow dynamics after appropriate averaging over the fast motion. The equation for slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

A new phase model of the spatiotemporal relationships between three circadian oscillators in the brainstem

Analysis of ex vivo Per2 bioluminescent rhythm previously recorded in the mouse dorsal vagal complex reveals a characteristic phase relationship between three distinct circadian oscillators. These signals represent core clock gene expression in the area postrema (AP), the nucleus of the solitary tract (NTS) and the ependymal cells surrounding the 4th ventricle (4Vep). Initially, the data suggests a consistent phasing in which the AP peaks first, followed shortly by the NTS, with the 4Vep peaking 8-9 h later. Wavelet analysis reveals that this pattern is not consistently maintained throughout a recording, however, the phase dynamics strongly imply that oscillator interactions are present. A simple phase model of the three oscillators is developed and it suggests that realistic phase dynamics occur between three model oscillators with coupling close to a synchronisation transition. The coupling topology suggests that the AP bidirectionally communicates phase information to the NTS and the 4Vep to synchronise the three structures. A comparison of the model with previous experimental manipulations demonstrates its feasibility to explain DVC circadian phasing. Finally, we show that simulating steadily decaying coupling improves the model's ability to capture experimental phase dynamics.

Ticking and talking in the brainstem satiety centre: Circadian timekeeping and interactions in the diet-sensitive clock of the dorsal vagal complex

The dorsal vagal complex (DVC) is a key hub for integrating blood-borne, central, and vagal ascending signals that convey important information on metabolic and homeostatic state. Research implicates the DVC in the termination of food intake and the transition to satiety, and consequently it is considered a brainstem satiety centre. In natural and laboratory settings, animals have distinct times of the day or circadian phases at which they prefer to eat, but if and how circadian signals affect DVC activity is not well understood. Here, we evaluate how intrinsic circadian signals regulate molecular and cellular activity in the area postrema (AP), nucleus of the solitary tract (NTS), and dorsal motor nucleus of the vagus (DMV) of the DVC. The hierarchy and potential interactions among these oscillators and their response to changes in diet are considered a simple framework in which to model these oscillators and their interactions is suggested. We propose possible functions of the DVC in the circadian control of feeding behaviour and speculate on future research directions including the translational value of knowledge of intrinsic circadian timekeeping the brainstem.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

An Optimal Time for Treatment-Predicting Circadian Time by Machine Learning and Mathematical Modelling

Tailoring medical interventions to a particular patient and pathology has been termed personalized medicine. The outcome of cancer treatments is improved when the intervention is timed in accordance with the patient's internal time. Yet, one challenge of personalized medicine is how to consider the biological time of the patient. Prerequisite for this so-called chronotherapy is an accurate characterization of the internal circadian time of the patient. As an alternative to time-consuming measurements in a sleep-laboratory, recent studies in chronobiology predict circadian time by applying machine learning approaches and mathematical modelling to easier accessible observables such as gene expression. Embedding these results into the mathematical dynamics between clock and cancer in mammals, we review the precision of predictions and the potential usage with respect to cancer treatment and discuss whether the patient's internal time and circadian observables, may provide an additional indication for individualized treatment timing. Besides the health improvement, timing treatment may imply financial advantages, by ameliorating side effects of treatments, thus reducing costs. Summarizing the advances of recent years, this review brings together the current clinical standard for measuring biological time, the general assessment of circadian rhythmicity, the usage of rhythmic variables to predict biological time and models of circadian rhythmicity.

Coupling of cell fate selection model enhances DNA damage response and may underlie BE phenomenon

Double-strand break-induced (DSB) cells send signal that induces DSBs in neighbour cells, resulting in the interaction among cells sharing the same medium. Since p53 network gives oscillatory response to DSBs, such interaction among cells could be modelled as an excitatory coupling of p53 network oscillators. This study proposes a plausible coupling model of three-mode two-dimensional oscillators, which models the p53-mediated cell fate selection in globally coupled DSB-induced cells. The coupled model consists of ATM and Wip1 proteins as variables. The coupling mechanism is realised through ATM variable via a mean-field modelling the bystander signal in the intercellular medium. Investigation of the model reveals that the coupling generates more sensitive DNA damage response by affecting cell fate selection. Additionally, the authors search for the cause-effect relationship between coupled p53 network oscillators and bystander effect (BE) endpoints. For this, they search for the possible values of uncertain parameters that may replicate BE experiments' results. At certain parametric regions, there is a correlation between the outcomes of cell fate and endpoints of BE, suggesting that the intercellular coupling of p53 network may manifest itself as the form of observed BEs.

Sleep Modelling across Physiological Levels

Sleep and circadian rhythms are regulated across multiple functional, spatial and temporal levels: from genes to networks of coupled neurons and glial cells, to large scale brain dynamics and behaviour. The dynamics at each of these levels are complex and the interaction between the levels is even more so, so research have mostly focused on interactions within the levels to understand the underlying mechanisms—the so-called reductionist approach. Mathematical models were developed to test theories of sleep regulation and guide new experiments at each of these levels and have become an integral part of the field. The advantage of modelling, however, is that it allows us to simulate and test the dynamics of complex biological systems and thus provides a tool to investigate the connections between the different levels and study the system as a whole. In this paper I review key models of sleep developed at different physiological levels and discuss the potential for an integrated systems biology approach for sleep regulation across these levels. I also highlight the necessity of building mechanistic connections between models of sleep and circadian rhythms across these levels.

The Times of Our Lives: Interaction Among Different Biological Periodicities

Environmental cycles on Earth display different periodicities, including daily, tidal or annual time scales. Virtually all living organisms have developed temporal mechanisms to adapt to such changes in environmental conditions. These biological timing structures—ranging from microsecond to seasonal timing—may have intrinsic properties and even different clock machinery. However, interaction among these temporal systems may present evolutionary advantages, for example, when species are exposed to changing climatic conditions or different geographic locations. Here, we present and discuss a model that accounts for the circadian regulation of both ultradian (less than 24-h) and infradian (more than 24-h) cycles and for the interaction among the three time scales. We show two clear examples of such interaction: (i) between the circadian clock and the seasonal regulation of the Hypothalamic-Pituitary-Thyroid (HPT) axis; and (ii) between the circadian clock and the hypothalamic-nigrostriatal (HNS) ultradian modulation. This remarkable interplay among the otherwise considered isolated rhythms has been demonstrated to exist in diverse organisms, suggesting an adaptive advantage of multiple scales of biological timing.

Neuronal oscillations on an ultra-slow timescale: daily rhythms in electrical activity and gene expression in the mammalian master circadian clockwork

Neuronal oscillations of the brain, such as those observed in the cortices and hippocampi of behaving animals and humans, span across wide frequency bands, from slow delta waves (0.1 Hz) to ultra-fast ripples (600 Hz). Here, we focus on ultra-slow neuronal oscillators in the hypothalamic suprachiasmatic nuclei (SCN), the master daily clock that operates on interlocking transcription-translation feedback loops to produce circadian rhythms in clock gene expression with a period of near 24 h (< 0.001 Hz). This intracellular molecular clock interacts with the cell's membrane through poorly understood mechanisms to drive the daily pattern in the electrical excitability of SCN neurons, exhibiting an up-state during the day and a down-state at night. In turn, the membrane activity feeds back to regulate the oscillatory activity of clock gene programs. In this review, we emphasise the circadian processes that drive daily electrical oscillations in SCN neurons, and highlight how mathematical modelling contributes to our increasing understanding of circadian rhythm generation, synchronisation and communication within this hypothalamic region and across other brain circuits.

Modeling and simulating the Arabidopsis thaliana circadian clock using XPP-AUTO

Circadian clocks are endogenous timekeepers that produce oscillations with a period of about one day. Their rhythmicity originates from complex gene regulatory networks at the cellular level. In the last decades, computational models have been proven to be a powerful tool in order to understand the dynamics and design principles of the complex regulatory circuitries underlying the circadian clocks of different organisms. We present the process of model development using a small and simplified two-gene regulatory network of the Arabidopsis circadian clock. Subsequently, we discuss important numerical techniques to analyze such a mathematical model using XPP-AUTO. We show how to solve deterministic and stochastic ordinary differential equations and how to compute bifurcation diagrams or simulate phase-shift experiments. We finally discuss the contributions of modeling to the understanding and dissection of the Arabidopsis circadian system.

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.